In the previous talk, we showed how to phrase the Galois-theoretic problem of constructing extensions of the rational numbers that ramify at a prescribed set of primes in terms of a question about Galois representations satisfying certain local conditions. In this talk, we will show how we can construct many such Galois representations by working on the automorphic side of the Langlands correspondence.

We construct the desired automorphic representations by applying the Arthur—Selberg trace formula and its variants. But in order to attach Galois representations to an automorphic representation of GL(n) with what is currently known, we need to ensure that the automorphic representation satisfies some delicate conditions. We will show how we overcome these difficulties by exploiting (1) the principle of twisted endoscopy in the Langlands program, (2) restrictions imposed by global self-duality for automorphic representations, and (3) equidistribution phenomena for families of automorphic representations varying in the weight aspect.